I wanted to mention the question of existence of full exceptional collections on toric Fano varieties, but this was answered in a very recent paper by Efimov. Probably there are still interesting questions left regarding derived categories of toric varieties...
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Piotr AchingerOct 27 '10 at 22:11

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Let $X$ be a complete toric variety, necessarily neither smooth nor projective. Is there a nontrivial vector bundle on $X$? Payne has constructed examples that have no one- or two-dimensional bundles, and constructing vector bundles of higher rank on these is still open. The "mirror" question is whether there exists a Lagrangian submanifold of $(\mathbb{C}^*)^n$ satisfying certain asymptotic conditions coming from the fan of $X$.
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David TreumannOct 28 '10 at 1:21

Although I don't really follow this, but I think that the following is an open conjecture. Conjecture Every ample divisor on a smooth toric variety is very ample and induces a projectively normal embedding. Is that right?
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Karl SchwedeOct 28 '10 at 4:04

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@Piotr. Having looked at the introduction to the Efimov paper, it seems that there are no new positive results there about full exceptional collections. Kawamata showed there is always a full exceptional collection of coherent sheaves on a smooth toric DM stack. And Orlov (says Efimov) conjectures that there should exist a strong full exceptional collection. It was conjectured that toric Fano varieties should admit full exceptional collections of line bundles. Efimov shows that this is false in general. Apologies to Efimov if I read him incorrectly.
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Chris BravOct 28 '10 at 7:59

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@Karl: ample $\Rightarrow$ very ample is known for smooth toric varieties, but ample $\Rightarrow$ projectively normal is indeed a well-known open question. (I say question'' rather than conjecture'' as it doesn't seem like all experts believe it.)
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Arend BayerOct 28 '10 at 12:34

Is the toric ideal of a smooth
projectively normal toric variety
generated by quadrics?

This is interesting, since toric ideals have an explicit description. In particular, it is not known if the coordinate ring of a smooth projectively normal toric variety is Koszul. Smoothness is of course essential here, since there are many toric hypersurfaces of degree $\ge 3$, e.g., $x_0^n=x_1 \cdots x_n$.